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On representing claims for coherent risk measures

Jacka, Saul D. and Berkaoui, Abdelkarem
(2007)
On representing claims for coherent risk measures.
Working Paper.
Coventry: University of Warwick. Centre for Research in Statistical Methodology.
Working papers, Vol.2007
(No.20).

Abstract

We consider the problem of representing claims for coherent risk measures.For this purpose we introduce the concept of (weak and strong) time-consistency withrespect to a portfolio of assets, generalizing the one defined in Delbaen [7].In a similar way we extend the notion of m-stability, by introducing weak and strongversions. We then prove that the two concepts of m- stability and time-consistencyare still equivalent, thus giving necessary and sufficient conditions for a coherent riskmeasure to be represented by a market with proportional transaction costs. We go onto deduce that, under a separability assumption, any coherent risk measure is stronglytime-consistent with respect to a suitably chosen countable portfolio, and show theconverse: that any market with proportional transaction costs is equivalent to a marketpriced by a coherent risk measure, essentially establishing the equivalence of the twoconcepts.